Direct proof

In mathematics and logic, a direct proof is a way of showing the truth or falsehood of a given statement by a straightforward combination of established facts, usually existing lemmas and theorems, without making any further assumptions. In order to directly prove a conditional statement of the form "If p, then q", it suffices to consider the situations in which the statement p is true. Logical deduction is employed to reason from assumptions to conclusion. The type of logic employed is almost invariably first-order logic, employing the quantifiers for all and there exists. Common proof rules used are modus ponens and universal instantiation.

In contrast, an indirect proof may begin with certain hypothetical scenarios and then proceed to eliminate the uncertainties in each of these scenarios until an inescapable conclusion is forced. For example instead of showing directly p ⇒ q, one proves its contrapositive ~q ⇒ ~p (one assumes ~q and shows that it leads to ~p). Since p ⇒ q and ~q ⇒ ~p are equivalent by the principle of transposition (see law of excluded middle), p ⇒ q is indirectly proved. Proof methods that are not direct include proof by contradiction. Direct proof methods include proof by exhaustion, proof by infinite descent, and proof by induction.

Example

What follows is a simple, direct proof that the sum of two even integers is itself an even number.

Consider two even integers x and y. Since they are even, they can be written as x=2a and y=2b respectively for integers a and b. Then the sum x + y = 2a + 2b = 2(a + b). From this it is clear x+y has 2 as a factor and therefore is even, so the sum of any two even integers is even.

References

• Franklin, J.; A. Daoud (2011). Proof in Mathematics: An Introduction. Sydney: Kew Books. ISBN 0646545094. http://www.maths.unsw.edu.au/~jim/proofs.html.  (Ch. 1.)

This logic-related article is a stub. You can help Wikipedia by expanding it.v · Categories: Proofs Logical truth Logic stubs Wikimedia Foundation. 2010. Игры ⚽ Поможем написать курсовую Paperback Avogadro's law Look at other dictionaries: Proof — • The establishment of a disputed or controverted matter by lawful means or arguments. Catholic Encyclopedia. Kevin Knight. 2006. Proof     Proof      …   Catholic encyclopedia direct evidence — see evidence Merriam Webster’s Dictionary of Law. Merriam Webster. 1996. direct evidence n …   Law dictionary Proof-of-payment — or POP is an honor based fare collection approach used on many public transportation systems. Instead of checking each passenger as they enter a fare control zone, proof of payment requires that each passenger carry a ticket or pass proving that… …   Wikipedia Direct method — may refer to Direct method (education) for learning a foreign language Direct method (computational mathematics) as opposed to iterative method Direct methods (crystallography) for estimating the phases of the Fourier transform of the scattering… …   Wikipedia direct evidence — Proof which speaks directly to the issue, requiring no support by other evidence; proof in testimony out of the witness own knowledge, as distinguished from evidence of circumstances from which inferences must be drawn if it is to have probative… …   Ballentine's law dictionary proof — The effect of evidence; the establishment of a fact by evidence. New England Newspaper Pub. Co. v. Bonner, C.C.A.Mass., 77 F.2d 915, 916. Any fact or circumstance which leads the mind to the affirmative or negative of any proposition. The… …   Black's law dictionary Proof-of-work system — A Proof of work ( POW ) system (or protocol, or function) is an economic measure to deter denial of service attacks and other service abuses such as spams on a network by requiring some work from the service requester, usually meaning processing… …   Wikipedia proof — noun ADJECTIVE ▪ clear, convincing, direct, good, positive, real ▪ |Do you have positive proof that she took the money? ▪ We have proof positive that he is in hid …   Collocations dictionary Direct Instruction — This article discusses a particular pedagogical model, as is found in DISTAR and related programs. For more general discussion see direct instruction. Direct Instruction (DI) is an instructional method that is focused on systematic curriculum… …   Wikipedia Direct demonstration — Demonstration Dem on*stra tion, n. [L. demonstratio: cf. F. d[ e]monstration.] 1. The act of demonstrating; an exhibition; proof; especially, proof beyond the possibility of doubt; indubitable evidence, to the senses or reason. [1913 Webster]… …   The Collaborative International Dictionary of English 18+ © Academic, 2000-2024 Contact us: Technical Support, Advertising Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx. Mark and share Search through all dictionaries Translate… Search Internet Share the article and excerpts Direct link … Do a right-click on the link above and select “Copy Link”